notebooks/11_probability.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "6a152a61",
   "metadata": {},
   "source": [
    "## Random and Pseudorandom Numbers\n",
    "\n",
    "Perhaps, the easiest way to generate bits that *look* random is to use the `randint` function from the `random` module in `Python`. The call `randint(a, b)` returns some integer in $\\{\\texttt{a},\\ldots,\\texttt{b}\\}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "a21f4c76",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "101101010110001111001010001101\n"
     ]
    }
   ],
   "source": [
    "from random import randint\n",
    "\n",
    "print(*[randint(0, 1) for _ in range(30)], sep='')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2a748739",
   "metadata": {},
   "source": [
    "If you run the same code again, you get different bits. However, these bits are obtained *not* by a random process but by some algorithm. This algorithm takes a *seed* and then generates a sequence of bits that depends on the seed in a rather complicated way — chosen in such a way that is difficult to distinguish this sequence from randomly generated bits (if you do not know the seed). \n",
    "\n",
    "We may specify the seed explicitly. Let us add a call `seed(29)` to our previous code."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "570841e0",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "011011100010101110100110111011\n"
     ]
    }
   ],
   "source": [
    "from random import randint, seed\n",
    "\n",
    "seed(29)\n",
    "print(*[randint(0, 1) for _ in range(30)], sep='')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6968f7f1",
   "metadata": {},
   "source": [
    "Then, each time you run this code, you get exactly the same thirty bits! This behavior of pseudorandom generators\n",
    "is useful in practice. If you debug some algorithm that uses random bits, and some error happens, you want the error to be reproducible. For that, you need to run the algorithm again with the same bits, and you achieve this by using the same seed.\n",
    "\n",
    "On the other hand, if you do not use this feature, the bit generator uses some seed that depends on time, so you will get new bits when you run the program again. There are good chances that you will never find out any difference between *pseudorandom* bits generated by `Python` and randomly generated bits when playing with simple programs, so we use pseudorandom bits a lot in this book (as if they were random) and encourage you to do the same — unless you need random bits for serious applications (especially, cryptographic ones)."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a9cd9d7e",
   "metadata": {},
   "source": [
    "## Frequency of Tails"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "940fec67",
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "from random import randint, seed\n",
    "\n",
    "# toss a coin many times\n",
    "seed(19)\n",
    "num_experiments = 1000\n",
    "tosses = [randint(0, 1) for _ in range(num_experiments)]\n",
    "frequencies = [sum(tosses[:i]) / i for i in range(1, num_experiments)]\n",
    "\n",
    "# plot\n",
    "plt.plot(frequencies)\n",
    "plt.ylabel(ylabel='Frequency of 1s')\n",
    "plt.xlabel(xlabel='Number of experiments')\n",
    "plt.ylim(bottom=0, top=1)\n",
    "plt.axhline(y=0.5, color='green')\n",
    "plt.savefig('freq.png')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5de6949b",
   "metadata": {},
   "source": [
    "## Galton Board"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "79643ee2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "from random import randint, seed\n",
    "\n",
    "seed(29)\n",
    "\n",
    "num_layers = 10\n",
    "num_beans = 3000\n",
    "\n",
    "\n",
    "def get_random_box():\n",
    "    return sum([randint(0, 1) for _ in range(num_layers)])\n",
    "\n",
    "\n",
    "counts = [0] * (num_layers + 1)\n",
    "for _ in range(num_beans):\n",
    "    counts[get_random_box()] += 1\n",
    "\n",
    "\n",
    "plt.bar(range(num_layers + 1), counts)\n",
    "plt.ylabel(ylabel='Number of beans')\n",
    "plt.xlabel(xlabel='Box number')\n",
    "plt.savefig('galton.png')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "331cb734",
   "metadata": {},
   "source": [
    "It is instructive to compare simulation results with our mathematical analysis. We computed the fraction of beans that should end in each box if the flow of beans is divided evenly at every splitting point.  The program below \n",
    "adds dots that show the result of this computation; the function `comb(num_layers, i)` from the `math` module computes the binomial coefficients (numbers from Pascal's triangle). One could see that the blue bars (simulation results) are quite close to them."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "45b37c75",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from math import comb\n",
    "import matplotlib.pyplot as plt\n",
    "from random import randint, seed\n",
    "\n",
    "seed(29)\n",
    "\n",
    "num_layers = 10\n",
    "num_beans = 3000\n",
    "\n",
    "\n",
    "def get_random_box():\n",
    "    return sum([randint(0, 1) for _ in range(num_layers)])\n",
    "\n",
    "\n",
    "counts = [0] * (num_layers + 1)\n",
    "for _ in range(num_beans):\n",
    "    counts[get_random_box()] += 1\n",
    "\n",
    "theory_counts = [num_beans * comb(num_layers, i) / 2 ** num_layers for i in range(num_layers + 1)]\n",
    "\n",
    "plt.bar(range(num_layers + 1), counts)\n",
    "plt.plot(theory_counts, 'o-', color='brown')\n",
    "plt.ylabel(ylabel='Number of beans')\n",
    "plt.xlabel(xlabel='Box number')\n",
    "plt.savefig('galton10-3000-2.png')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9ec3f089",
   "metadata": {},
   "source": [
    "## Two Problems\n",
    "\n",
    "**Problem.** Six people, including Alice, Bob, and Charlie, form a queue in a random order.\n",
    " * What is the probability of the event \"Alice is the first in the queue\"?\n",
    " * What is the probability of the event \"Alice precedes Bob in the queue\"? (This means that Alice is closer to the start of the queue than Bob; there may be other people between them, or before/after them)\n",
    " * What is the probability of the event \"Bob is between Alice and Charlie in the queue\"? (Alice may be before or after Charlie, and Alice and Charlie may not be next to Bob.)\n",
    " * What is the probability of the event \"Alice and Bob are neighbors in the queue\"? (This means that Alice stays right before Bob or just after Bob)."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4f11ca8c",
   "metadata": {},
   "source": [
    "**Problem.** There are two boxes and six balls labeled $0,1,2,3,4,5$. The balls $0$ and $1$ are placed in one box and the balls $2,3,4,5$ are placed in the other box. You choose randomly one of the boxes (they look identical) and take a random ball from this box. Which ball has more chances to be chosen: $1$ or $4$?\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "9983b660",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "from random import randint, seed\n",
    "\n",
    "seed(27)\n",
    "\n",
    "num_experiments = 100\n",
    "count = [0] * 6\n",
    "\n",
    "for _ in range(num_experiments):\n",
    "    box = randint(0, 1)\n",
    "    if box == 0:\n",
    "        ball = randint(0, 1)\n",
    "    else:\n",
    "        ball = randint(2, 5)\n",
    "    count[ball] += 1\n",
    "\n",
    "plt.bar(range(6), count)\n",
    "plt.savefig('twourns.png')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "532e438e",
   "metadata": {},
   "source": [
    "**Problem.** Take three random points on a plane. What is the probability that they form (are vertices of) an acute triangle? (Acute triangle has all angles less than $90^\\circ$; right triangle has one angle $90^\\circ$, and obtuse triangle has one angle greater than $90^\\circ$.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "e5a4c071",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from random import random, seed\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "seed(37)\n",
    "\n",
    "\n",
    "def distance_squared(a, b):\n",
    "    return (a[0] - b[0]) ** 2 + (a[1] - b[1]) ** 2\n",
    "\n",
    "\n",
    "def acute(a, b, c):\n",
    "    sides_squared = [\n",
    "        distance_squared(a, b),\n",
    "        distance_squared(a, c),\n",
    "        distance_squared(b, c)\n",
    "    ]\n",
    "    sides_squared = sorted(sides_squared)\n",
    "    return sides_squared[0] + sides_squared[1] > sides_squared[2]\n",
    "\n",
    "\n",
    "def random_point(width, height):\n",
    "    return [random() * width, random() * height]\n",
    "\n",
    "\n",
    "def frequency_acute(width, height, num_experiments):\n",
    "    experiments = [1 if acute(\n",
    "        random_point(width, height),\n",
    "        random_point(width, height),\n",
    "        random_point(width, height)) else 0\n",
    "        for _ in range(num_experiments)\n",
    "    ]\n",
    "\n",
    "    return [sum(experiments[:i]) / i for i in range(1, num_experiments)]\n",
    "\n",
    "\n",
    "plt.plot(frequency_acute(100, 100, 10000), color='green')\n",
    "plt.plot(frequency_acute(210, 297, 10000), color='red')\n",
    "plt.plot(frequency_acute(100, 500, 10000), color='blue')\n",
    "plt.ylabel(ylabel='Frequency of acute triangles')\n",
    "plt.xlabel(xlabel='Number of experiments')\n",
    "plt.ylim(bottom=0, top=1)\n",
    "plt.savefig('freq_acute.png')        \n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7b1bad1f",
   "metadata": {},
   "source": [
    "## Monty Hall Paradox"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "473f0ce1",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.33161\n",
      "0.4991\n",
      "0.66682\n"
     ]
    }
   ],
   "source": [
    "from random import randint, seed\n",
    "\n",
    "seed(96)\n",
    "\n",
    "\n",
    "def initial_choice():\n",
    "    return randint(1, 3)\n",
    "\n",
    "\n",
    "def keep(choice, opened):\n",
    "    return choice\n",
    "\n",
    "\n",
    "def randomize(choice, opened):\n",
    "    rooms = [1, 2, 3]\n",
    "    rooms.remove(opened)\n",
    "    return rooms[randint(0, 1)]\n",
    "\n",
    "\n",
    "def change(choice, opened):\n",
    "    return 6 - choice - opened\n",
    "\n",
    "\n",
    "def success_frequency(strategy, num_experiments):\n",
    "    num_wins = 0\n",
    "    for _ in range(num_experiments):\n",
    "        car = randint(1, 3)\n",
    "        choice = initial_choice()\n",
    "\n",
    "        if choice != car:\n",
    "            opened = 6 - choice - car  # remaining\n",
    "        else: # car = choice\n",
    "            rooms = [1, 2, 3]\n",
    "            rooms.remove(car)\n",
    "            opened = rooms[randint(0, 1)]\n",
    "\n",
    "        new_choice = strategy(choice, opened)\n",
    "\n",
    "        if new_choice == car:\n",
    "            num_wins += 1\n",
    "\n",
    "    return num_wins / num_experiments\n",
    "\n",
    "\n",
    "print(success_frequency(keep, 100000))\n",
    "print(success_frequency(randomize, 100000))\n",
    "print(success_frequency(change, 100000))\n"
   ]
  }
 ],
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  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.9.13"
  }
 },
 "nbformat": 4,
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}